# On*s*-sets and mutual absolute continuity of measures on homogeneous spaces

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## Abstract

We extend a recent result of A. Jonsson about mutual absolute continuity of two*D* _{ s }-measures on an*s*-set*F* ⊂*R* ^{ n } to the homogeneous spaces (*X, d, μ*) of Coifman, Weiss. Here we define Hausdorff measure, Hausdorff dimension,*D* _{ s }-set and*d*-set relative to the measure*μ*. Our main result holds for so called (*s, d*)-sets,*d* ≥*s*, and is stronger than Jonssons result even in*R* ^{ n }. As applications we interpret this Hausdorff dimension as a relative dimension for very regular sets and show that it in general depends strongly on*μ*. For this purpose we construct a strictly increasing function*f* :*R* →*R*, whose measure is doubling and concentrated on a set of arbitrary small Hausdorff dimension. The extension of*f* to a quasiconformal map of the half plane onto itself sharpens a classical example of Ahlfors-Beurling.

## Keywords

Homogeneous Space Hausdorff Dimension Besov Space Quasiconformal Mapping Hausdorff Measure## Preview

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